For the rice: Place the rice in a strainer and rinse until the water runs clear. Drain well. Combine the rice and water in a rice cooker and cook for 30 minutes. Meanwhile, in a small saucepan, combine the vinegar, oil, sugar and salt and cook over medium heat, stirring, just until the salt and sugar dissolve. When the rice is done, dump it into a hangiri (wooden tub) or into a large mixing bowl and pour the vinegar mixture over top. Spread and cut the vinegar into the rice evenly with a rice paddle or wooden spatula. Let the rice cool for 10 minutes, then flip the rice over and cool another 10 minutes.
For the sauce: Combine all the sauce ingredients in a bowl and blend with an immersion blender or whisk. Cover and refrigerate.
For the sushi: Heat the oil in a deep fryer or large heavy skillet until a deep-fry thermometer registers 350 degrees F. Place the eggs in a shallow bowl. Place the flour on a plate and the panko on another plate. One at a time, dip a crab in the flour, then in the egg wash and then roll in the panko. Fry the crab for about 2 minutes on each side. Then place the crab on paper towels to cool. Repeat with the remaining crabs.
Place a sheet of nori on a makisu (bamboo sushi mat) and cover the nori with 3 to 4 ounces of the cooled sushi rice. Place one handful kaiware, 3 slices avocado, 1 piece yamagobo and 1 stick kani across the roll. Cut a crab into 4 pieces. Put the legs on the ends so they stick out over the ends of the nori. Place the rest of the crab in the middle. Use the makisu (bamboo sushi mat) to help you roll the nori and rice tightly around the filling ingredients. Repeat with the remaining sushi ingredients and rice, making 4 rolls.
Cut the rolls in half and enjoy them dipped into the sauce.